The 26-year-old has been strongly linked with a move to the Etihad Stadium, with Pep Guardiola identifying the England international as his top target to replace Fernandinho.
Despite interest from a host of other Premier League clubs including the likes of Manchester United, Liverpool, Aston Villa and West Ham United, as well as French giants Paris Saint-Germain, Phillips is understood to have told Leeds that he would like to join Man City this summer.
According to The Athletic and the Daily Telegraph, the Premier League champions are set to pay an initial £42m fee to sign Phillips, which could rise to £45m with add-ons.
Both reports add that Man City's 18-year-old academy graduate Darko Gyabi is to move to Elland Road in a deal worth £5m.
Phillips is currently on holiday and is due to report back for pre-season training with Leeds on July 4, but progress in negotiations with the Citizens could see the deal finalised as early as next week.
The Yorkshireman has spent the last 12 years with his boyhood club since joining the academy in 2010, and he has made a total of 233 appearances across all competitions.
As soon as Man City complete the signing of Phillips, Guardiola's side is set to turn their attention to Brighton & Hove Albion left-back Marc Cucurella, who is valued in the region of £50m by the Seagulls.